¶ Wind Triangle Derivation
For aviation and nautical navigation, the wind triangle is very important for determining the relations between the course and heading. Here, the general wind triangle is solved with an aviation flavor.
The wind triangle can be solved via permutations of the two governing geometric relationships derived from the Law of Sines and Law of Cosines (notation is explained in later):
These equations hold for the domain of angles between . These equations assume that the wind speed are approximately less than of the true air speed. They work at higher ratios with one caveat.
¶ Overview
The wind triangle is generally used to solve 1 of 3 different problems, two of which is considered the "direct" problem and the last being the "inverse" problem. Nihad E. Daidzic 2016
- Direct Problems:
- The airspeed, heading, wind direction, and wind speed are known; the ground speed, drift angle, and the true course are calculated. (This problem shall be called the secondary direct problem.)
- The airspeed, true course, wind direction, and wind speed are known; the ground speed, wind correction angle, and required heading are calculated. (This problem shall be called the primary direct problem.)
- Inverse Problem:
- The airspeed, ground speed, heading, and course are known. The wind direction and wind speed are calculated. (This problem shall be called the primary inverse problem.)
The solutions to these problems in generalized mathematical form are provided.
¶ Notation and Primary Diagram
We will use the following wind triangle as a visual and notional reference along with the following notation: (For the sake of the equations often found on a E6B, , the drift angle is and the wind correction angle is .)
(Only the physically important angles are listed.)
Notation:
| Notation | Quantity |
|---|---|
| V | Aircraft airspeed |
| G | Aircraft groundspeed |
| W | Wind speed |
| Aircraft heading | |
| Aircraft course or track | |
| Wind direction | |
| Drift angle (Wind correction angle ) |
¶ Geometric and Trigonometric Identities
Three separate geometric and trigonometric identities are needed. Consider a triangle with the following general dimensions:
Euclidean triangles such as these follow these properties:
- Triangle Postulate The sum of all angles of the triangle is equal to two right angles. Thus:
- Law of Sines The ratio of sides and the sines of their opposite angles are equal. Thus:
- Law of Cosines The Pythagorean theorem extended for any triangle that is not a right triangle. Thus:
Two other miscellaneous geometric properties are needed.
- Supplementary Angles A linear pair of angles (two angles which share only one side and their non-shared sides are the same line) sum to a straight angle .
- Conjugate Angles A pair of non-congruent angles which share the same sides sum to a complete angle .
¶ Simple Angle Relationships
There exists many simple angle relationships.
¶ Supplementary and Conjugate Angles
The angles and and supplementary to each other, as such:
The angles and are conjugate angles. (Notice that both angles opposite to each other, these angles are congruent.) As such,
¶ Triangle Postulate Relationships
The angles , , and make up a single triangle, and, via the Triangle Postulate:
Similarly, the angles , , and make up a single triangle, and, via the Triangle Postulate:
Similarly, the angles , , and make up a single triangle, and, via the Triangle Postulate:
¶ Angle Addition Relationships
Angle addition also allows us to establish a relationship with , , and :
¶ Trigonometric Relationships
Using the law of sines on the primary wind triangle:
Using the law of cosines on the primary wind triangle:
¶ Aerodynamic Relationships
Often E6Bs are used for the quick slide-rule calculations of many of these values here. For these calculations, we use the drift angle defined by: (For the course and the heading .)
But, E6Bs have the notation: (Here, notationally, the wind correctional angle is given as .)
Conventionally, it is assumed that . However, this is only the case for where wind speed to airspeed ratio , the deviation being on the order of (Alexander W S et. al. 1941). That is assumed here, as such:
Should these equations be used in the domain of , corrections to the current course may need to be done more frequently to fix the small deviation.
¶ Governing Equations
Here, the equations governing the wind triangle quantities are derived. The primary means are formulations of the law of sines and law of cosines.
¶ Sines Equation
For the angle: We have, , thus . Using the previous Triangle Postulate :
And, given that :
For the angle: We have and ,
For the angle: We have and the previous relationships, so: (Of course, this should be consistent with from Angle Addition,
¶ Full Law of Sines
As such, the full Law of Sines can be as follows. The second conversion is because and .
¶ Cosines Equation
For the Angle: From Angle Addition:
This value, , can be substituted into the Law of Cosines, .
¶ Full Law of Cosines
Provided that , the full law of cosines is:
Two other equivalent law of cosines are:
¶ Solving the Wind Triangle
Here we present the general solutions to the different types of problems initally described, provided the governing equations found above.
Note, generally, the Law of Cosines is more numerically stable to rounding errors than the Law of Sines. It also does not suffer from the Law of Sines ambiguity.
¶ Primary Direct Problem
The following are known parameters:
- True airspeed
- True/magnetic course
- True/magnetic wind direction
- Wind speed
The following are unknown parameters and thus must be calculated:
- Ground speed
- Wind correction angle
- True/magnetic heading
Via the Law of Sines from the governing equations, the wind correction angle can be determined:
Via the Law of Sines from the governing equations, the heading can be determined:
Via the Law of Cosines from the governing equations, the ground speed can be determined:
¶ Secondary Direct Problem
The following are known parameters:
- True airspeed
- True/magnetic heading
- True/magnetic wind direction
- Wind speed
The following are unknown parameters and thus must be calculated:
- Ground speed
- Drift angle
- True/magnetic course
Via the Law of Cosines from the governing equations, the ground speed can be determined:
Via the Law of Sines from the governing equations, the course can be determined:
Via the Law of Sines from the governing equations, the drift angle can be determined:
¶ Primary Inverse Problem
The following are known parameters:
- True airspeed
- Ground speed
- Heading
- Course
The following are unknown parameters and thus must be calculated:
- Wind direction
- Wind speed
Via the Law of Cosines from the governing equation, the wind speed can be determined:
Via the Law of Sines from the governing equation, the wind direction can be determined: